Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ 1+2 x+\frac{2^{2} x^{2}}{2 !}+\frac{2^{3} x^{3}}{3 !}+\frac{2^{4} x^{4}}{4 !}+\cdots $$

Step-by-Step Solution

Answer

The power series converges for all real numbers; it has an infinite radius of convergence.

Step 1: Identifying the nth Term

The given series is a power series, and we need to find an explicit formula for the nth term. Observing the terms, we see a pattern where each term involves \(2^n x^n / n!\). Therefore, the nth term \(a_n\) can be expressed as \(a_n = \frac{2^n x^n}{n!}\).

Step 2: Applying the Absolute Ratio Test

The Absolute Ratio Test involves checking the limit: \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). Substituting the nth term formula, we get \[ \frac{a_{n+1}}{a_n} = \frac{2^{n+1} x^{n+1} / (n+1)!}{2^n x^n / n!} = \frac{2 x}{n+1}. \] We need to evaluate \( \lim_{n \to \infty} \left| \frac{2x}{n+1} \right| \).

Step 3: Calculating the Limit of the Ratio

Simplify the expression \( \left| \frac{2x}{n+1} \right| \). As \(n\) approaches infinity, \(\frac{1}{n+1}\) approaches zero, implying that \(\lim_{n \to \infty} \left| \frac{2x}{n+1} \right| = 0\).

Step 4: Conclusion on Convergence

Since the limit approached zero for any real number (ignoring the value of \(x\))—this means the series converges for all \(x\). The radius of convergence is therefore infinite.

Key Concepts

Absolute Ratio Test

The Absolute Ratio Test is a useful tool for determining the convergence of a power series. It involves checking the limit of the ratio of consecutive terms. For a power series with a general term denoted as \(a_n\), we consider the expression
\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]If this limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. The test is inconclusive if the limit equals 1.

• This test helps determine where the series converges based on the variable \(x\).
• It gives insight into the behavior of the series as \(n\) grows larger.

Simply substituting and simplifying the formula for \(a_n = \frac{2^n x^n}{n!}\), we found
\[\frac{2 x}{n+1}\]as the ratio for successive terms. As \(n\) approaches infinity, this ratio approaches zero, indicating convergence for any real \(x\). Thus, the power series converges absolutely for all values of \(x\).

radius of convergence

The radius of convergence informs us about the set of \(x\) values for which a power series converges. In the context of the Absolute Ratio Test used earlier, the radius can be identified by seeking the range of values that result in convergence.

• When the limit of the ratio is zero, the radius of convergence is said to be infinite.
• In this case, it means the series converges for every real number \(x\).

For our problem, since the limit of the ratio expression \(\left| \frac{2x}{n+1} \right|\) approaches zero regardless of \(x\), the power series converges for all real numbers. Therefore, the radius of convergence is infinite. This indicates that the series forms a function defined across the entire real line.

nth term formulation

Determining the formula for the \(n\)th term is crucial in analyzing a power series. The \(n\)th term formulation allows us to utilize various convergence tests. In the given series, we observe the pattern
\[1, 2x, \frac{2^2x^2}{2!}, \frac{2^3x^3}{3!}, \ldots\]From this, the general term is identified as
\[a_n = \frac{2^n x^n}{n!}\]This representation is vital for applying the Ratio Test, as it provides the iterative elements necessary to explore the limits of the converging sequence.

• The \(n\)th term lets us establish the parameter that varies with \(x\).
• It forms the basis for mathematical manipulations and further calculations.

Finding this formula at the start helps streamline the entire process of determining convergence characteristics. By understanding the structure of the \(n\)th term, we gain insights that lead to assessing convergence and identifying the radius of convergence.